Estimate of an Hypoelliptic Heat-Kernel outside the Cut-Locus in Semi-Group Theory

We give a proof in semi-group
theory based on the Malliavin Calculus of Bismut type in semi-group theory and
Wentzel-Freidlin estimates in semi-group of our result giving an expansion of
an hypoelliptic heat-kernel outside the cut-locus where Bismut’s non-degeneray condition plays a
preominent role.